Intonation is about math. Many people feel that it can be “expressive” in that a solo line can be inflected subtly to exaggerate harmonic gravity, but in ensembles, it is all about the math of sum and difference tones. For me, this would include playing solo, as our instruments have natural resonances that can be enhanced by intonation that exploits it. Intonation is dependent on which combination of notes is played, because the pleasing set of combined tones (not just the notes played but their interactions) is what we like. Exploiting it can also increase resonance and projection.
Sum and difference frequencies:
When you play a note, you are shortening the string. This is due to the physics of a string, the lower the note, the longer the string, the higher the note, the shorter the string. It is not linear. A piano has the strings set to particular pitches and you push buttons that are equally spaced apart. If you look at the frets on a guitar, you can see that they are closer together the higher the pitches are.
If you feel that you are uncertain where notes are, you need to spend time with scales, slow scales, so you can really pay attention to what is going on with your body and the instrument. Eventually your mental map of the fingerboard improves. It can be very helpful to play drone open strings with your slow scale work, ie, when you finger notes on the C string, play them together with the open G, when you play on the G string-open D, when you play on the D string, open A, and when you play of the A string, play open D. Of course, the sum and difference tones will be somewhat distorted depending on what combinations you are playing, but it is a reasonable way to work on the mental map.
Learning the notes, as Starker often remarked, is a matter of learning the map of the fingerboard. By this he meant a mental map of the fingerboard. Once you understand a position, it is mapped in your mind. I don’t think this is especially visual, as you need to understand how intervals relate to one another in a position. This is especially true in thumb position. Once you understand the tonal relationships accessible within a position, training the balance of the hand to maximize that access and minimize tension is feasible. So it really comes down to a combination of mental mapping of the fingerboard and ear training.
I would add that there are no muscles in the fingers, they are controlled entirely by tendons that are attached to muscles that are attached at the elbow. To maintain the position of the hand in the “holding an apple” metaphor takes a small amount of strength, and it should feel reasonably relaxed to do so. if the position of the hand and fingers is good, then a slight rotation of the forearm accomplishes most of what is required to put the fingers down and pick them up, a small flick of the tendons and it is complete. Squeezing and putting fully down more fingers than the finger required for the current note only adds tension and compromises control. It takes a good deal of time to master these things with consistency, so be patient. Scales are an excellent way to build this, as there is little to distract one’s focus from the technique.
When the arm approaches the neck at the correct angle to use rotation/leverage of the forearm to shift the weight of the arm and hand into the finger that is playing, it takes very little pressure to stop the string. The wrist should always be in neutral position, which lowers tension on the tendons that control the fingers where they pass through the wrist. The base knuckles of the hand should be arched such that the palm and thumb do not collapse. look at the 2 minute mark in the video below (which is about bow technique, but you will see how the fingers should work best in upper positions)
neutral position of wrist
Lynn Harrell on left hand pressure
There are a number of factors that effect the placement of fingers for intonation. The typical height of the A above the end of the fingerboard is 5.5mm, and the C is 8.5mm. The higher the pitch of the string, the shorter its transit is, the lower the pitch, the greater its transit. Transit means the complete distance the string travels in a single cycle that produces its pitch. Because of the difference in transit, all of the strings are a different height above the fingerboard, which means that if you press them down to the fingerboard at exactly the same distance from the nut on the different strings, you will distort the pitch of the highest string less than the lowest string, meaning the fifths will be out of tune (the lower the string, the sharper it will be). In addition, each of your finger pads is a different shape and thickness, so this also influences what happens when you put your fingers down (especially if you are stopping a fifth with one finger). So many complications…
Many inexpensive instruments have necks (and more commonly fingerboards) that flex and twist as you play, which really makes intonation a moving target. This can be addressed, as if the fingerboard is not ebony, that can be changed, which will stiffen the neck, and some makers these days put a carbon fiber rod in the neck to stiffen it and add resonance.
There are strings that respond to changes in bow pressure and speed with changes in pitch more than others. In general, higher tension strings of smaller gauge resist this change in pitch much better that lower tension, thicker gauge strings do. They are also much more responsive to articulation. Refer to my post on strings for more information on this topic.
I have used a variety of tuner apps with different settings, and I think it is a mistake to obsess on the ?cents? too much, a strobe setting is sometimes less disruptive. I usually tune the open A to the tuner, then tune the rest by ear (those sum and difference tones!). Some instruments seem more sensitive than others in that if you tune all the strings one by one, you might find that the bridge and tailpiece fluctuations that result from the string movements might alter the original A, so it might take a little time to get it all balanced.